Were they Celtic, Germanic, neither? Were they sedentary or nomadic? In general how did they compare to the people living in Gaul on the other side of the channel?
Invasions (there were two of them) What do we know? A fair bit. Barry Cunliffe's Iron age Communities in Britain provides an overview of this period. Its also a bit over an inch thick. So lets stick with the more specific questions. Also I'm going to stick to England because Scotland (northern Scotland in particular) has some oddities.
Were they Celtic, Germanic, neither?
Culturally they were Celtic. We see this in their coins (mostly some have direct rather than indirect Greek influence) their grave goods, their weapons and their jewellery (torcs in particular . However they weren't unified and you can see cultural differences between say the Iceni and say the Durotriges. Even those cultural groups can be rather messy. While the Iceni seem to be a straightforward fairly unified group the nature of the Durotriges (even the name is poorly attested) is less well understood.
Were they sedentary or nomadic?
Sedentary. However we don't have an entirely solid grasp on the patterns of living. In particular the interaction between living in hillforts and living outside them. The roman invasion though does seem to have have resulted in some re-fortification being done on some hillforts. Static shires like Hayling Island also existed.
In general how did they compare to the people living in Gaul on the other side of the channel?
Less Romanised? There were strong trade links between Britian and Gaul and the disruption caused by Caesar's conquest of Gaul resulted in political instability in southern Britian
Being any more precise than that runs you into the problem of the Belgae invasion. The Belgae were a group in northern Gaul who Caesar claimed invaded and conquered part of Britain. The archaeological evidence however is unclear and this claim is disputed. So its not clear if an invasion happened and if it did did it involve a large scale movement of people or just a limited tribal elite. So depending on your position you can argue and britian and gaul in places contained literaly the same people, had some elites in common or simply gaul having a heavy cultural influence.
Sources.
Barry Cunliffe's Iron age Communities in Britain is pretty much the starting point for anything about Celtic Britian.
If you want something smaller Martin Papworth's The Search for the Durotriges has the advantage of covering a smaller geographical area allowing it to be slightly more reasonable in size.
The aside on coinage comes from Philip De Jersey's Celtic Coinage in Britain but thats not worth reading unless you want a short work on British Celtic coinage.
Not a source but worth mentioning.
If you want to learn about how people were living there is Butser Ancient Farm a longstanding experimental archaeology/ educational project:
They were all Celtic from what we can tell. Caesar refers to the Britons living opposite France as being similar in look to the Gauls and this also corresponds with coin finds being restricted to those eastern/southern tribes. They were definitely part of the wider Celtic/Gallic world.
Two things stand out. Mercenaries from Briton were sent to Gaul to help in the fight against Caesar and there were political, maybe family ties that joined the two regions together. The Druid centres in Britain (Anglesey?) were regarded as the most prestigious places for druids in Gaul to go learn.
Overall the Britons appear to be somewhat more archaic than the ones in Gaul. They relied on chariots in warfare, some at least appear to have dyed their skin before battle, they appear to have lacked the amount of armour seen in Gallic graves.
The northern Britons used smaller shields, a design that is still being used centuries later by the Picts and which also appear in Ireland. Things which tend to get ignored. The Britons had harbours with cargo ships involved in cross channel trade. We hear lots about little coracles, but next to nothing about these ships with heavy sails, high prows, likely similar to a medieval cog.
They had roads. Not paved, but roads. Boggy terrain had wooden planks laid down on piles. Exports noted by the Romans were copper, tin, large dogs (like wolfhounds), wheat and a heavy winter cloak which was a regional peculiarity that was popular amongst well to do Romans.